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Drilling Systems: Stability and Hidden Oscillations

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Discontinuity and Complexity in Nonlinear Physical Systems

Part of the book series: Nonlinear Systems and Complexity ((NSCH,volume 6))

Abstract

There are many mathematical models of drilling systems Despite, huge efforts in constructing models that would allow for precise analysis, drilling systems, still experience breakdowns. Due to complexity of systems, engineers mostly use numerical analysis, which may lead to unreliable results. Nowadays, advances in computer engineering allow for simulations of complex dynamical systems in order to obtain information on the behavior of their trajectories. However, this simple approach based on construction of trajectories using numerical integration of differential equations describing dynamical systems turned out to be quite limited for investigation of stability and oscillations of these systems. This issue is very crucial in applied research; for example, as stated in Lauvdal et al. (Proceedings of the IEEE control and decision conference, 1997) the following phrase: “Since stability in simulations does not imply stability of the physical control system (an example is the crash of the YF22) stronger theoretical understanding is required”. In this work, firstly a mathematical model of a drilling system developed by a group of scientists from the University of Eindhoven will be considered. Then a mathematical model of a drilling system with perfectly rigid drill-string actuated by induction motor will be analytically and numerically studied. A modification of the first two models will be considered and it will be shown that even in such simple models of drilling systems complex effects such as hidden oscillations may appear, which are hard to find by standard computational procedures.

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Notes

  1. 1.

    In the 1950–1960s of last century the investigations of widely known Markus-Yamabe’s, Aizerman’s conjecture (Aizerman problem), and Kalman’s conjecture (Kalman problem) on absolute stability led to the finding of hidden oscillations in automatic control systems with nonlinearity, which belongs to the sector of linear stability (see, e.g., [2, 17, 23, 27, 29] and others). In 1961, Gubar’ [7] showed analytically the possibility of hidden oscillations existence in two-dimensional system of phase locked-loop [32, 33]. In 2010 chaotic hidden oscillations (hidden attractors) were discovered for the first time [15, 16, 28, 30, 31] in Chua’s circuit.

References

  1. Al-Bender F, Lampaert V, Swevers J (2004) Modeling of dry sliding friction dynamics: From heuristic models to physically motivated models and back. Chaos 14(2):446–460

    Article  Google Scholar 

  2. Bragin VO, Vagaitsev VI, Kuznetsov NV, Leonov GA (2011) Algorithms for finding hidden oscillations in nonlinear systems. The Aizerman and Kalman conjectures and Chua’s circuits. J Comput Syst Sci Int 50(4):511–543, DOI 10.1134/S106423071104006X

    MathSciNet  MATH  Google Scholar 

  3. Brett J (1992) Genesis of torsional drillstring vibrations. SPE Drilling Eng 7(3):168–174

    Google Scholar 

  4. Brockley C, Cameron R, Potter A (1967) Friction-induced vibrations. ASME J Lubricat Technol 89:101–108

    Article  Google Scholar 

  5. de Bruin J, Doris A, van de Wouw N, Heemels W, Nijmeijer H (2009) Control of mechanical motion systems with non-collocation of actuation and friction: a Popov criterion approach for input-to-state stability and set-valued nonlinearities. Automatica 45(2):405–415

    Google Scholar 

  6. Filippov AF (1988) Differential equations with discontinuous right-hand sides. Kluwer, Dordrecht

    Book  MATH  Google Scholar 

  7. Gubar’ NA (1961) Investigation of a piecewise linear dynamical system with three parameters. J Appl Math Mech 25(6):1011–1023

    MathSciNet  Google Scholar 

  8. Horbeek J, Birch W (1995) In: Proceedings of the society of petroleum engineers offshore, Europe, pp 43–51

    Google Scholar 

  9. Ibrahim R (1994) Friction-induced vibration, chatter, squeal, and chaos: dynamics and modeling. Appl Mech Rev: ASME 47(7):227–253

    Article  Google Scholar 

  10. Ivanov-Smolensky A (1980) Electrical machines. Energiya, Moscow

    Google Scholar 

  11. Jansen J (1991) Non-linear rotor dynamics as applied to oilwell drillstring vibrations. J Sound Vibration 147(1):115–135

    Article  Google Scholar 

  12. Kiseleva MA, Kuznetsov NV, Leonov GA, Neittaanmäki P (2012) Drilling systems failures and hidden oscillations. In: IEEE 4th international conference on nonlinear science and complexity, NSC 2012 – Proceedings, pp 109–112, DOI 10.1109/NSC.2012.6304736 http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6304736

  13. Kondrat’eva N, Leonov G, Rodjukov F, Shepeljavyj A (2001) Nonlocal analysis of differential equation of induction motors. Technische Mechanik 21(1):75–86

    Google Scholar 

  14. Kreuzer E, Kust O (1996) Analyse selbsterregter drehschwingugnen in torsionsstäben. ZAMM – J Appl Math Mech 76(10):547–557

    Article  MATH  Google Scholar 

  15. Kuznetsov N, Kuznetsova O, Leonov G, Vagaitsev V (2013) Informatics in control, automation and robotics. Lecture notes in electrical engineering, vol 174, Part 4, Chap. Analytical-numerical localization of hidden attractor in electrical Chua’s circuit. Springer, Berlin, pp 149–158. DOI 10.1007/978-3-642-31353-0∖_11

    Google Scholar 

  16. Kuznetsov NV, Leonov GA, Vagaitsev VI (2010) Analytical-numerical method for attractor localization of generalized Chua’s system. IFAC Proc Vol (IFAC-PapersOnline) 4(1):29–33, DOI 10.3182/20100826-3-TR-4016.00009

    Google Scholar 

  17. Kuznetsov NV, Leonov GA, Seledzhi SM (2011) Hidden oscillations in nonlinear control systems. IFAC Proc Vol (IFAC-PapersOnline) 18(1):2506–2510, DOI 10.3182/20110828-6-IT-1002.03316

    Google Scholar 

  18. Lauvdal T, Murray R, Fossen T (1997) Stabilization of integrator chains in the presence of magnitude and rate saturations: a gain scheduling approach. In: Proceedings of the 36th IEEE Conference on Decision and Control, Vol. 4, pp 4404–4005, DOI 10.1109/CDC.1997.652491

    Google Scholar 

  19. Leine R (2000) Bifurcations in discontinuous mechanical systems of filippov-type. Ph.D. thesis, Eindhoven University of Technology, The Netherlands

    Google Scholar 

  20. Leine R, Campen DV, Keultjes W (2003) Stick-slip whirl interraction in drillstring dynamics. ASME J Vibrat Acoustics 124

    Google Scholar 

  21. Leonov G, Kuznetsov N (2013) IWCFTA2012 Keynote Speech I - Hidden attractors in dynamical systems: From hidden oscillation in Hilbert-Kolmogorov, Aizerman and Kalman problems to hidden chaotic attractor in Chua circuits. In: 2012 Fifth International Workshop on Chaos-Fractals theories and applications (IWCFTA), pp XV–XVII, DOI 10.1109/IWCFTA.2012.8

    Google Scholar 

  22. Leonov GA, Kiseleva MA (2012) Analysis of friction-induced limit cycling in an experimental drill-string system. Doklady Phys 57(5):206–209

    Article  Google Scholar 

  23. Leonov GA, Kuznetsov NV (2011) Algorithms for searching for hidden oscillations in the Aizerman and Kalman problems. Doklady Math 84(1):475–481, DOI 10.1134/S1064562411040120

    Article  MathSciNet  MATH  Google Scholar 

  24. Leonov GA, Kuznetsov NV (2011) Analytical-numerical methods for investigation of hidden oscillations in nonlinear control systems. IFAC Proc Vol (IFAC-PapersOnline) 18(1): 2494–2505, DOI 10.3182/ 20110828-6-IT-1002.03315

    Google Scholar 

  25. Leonov GA, SolovTeva EP (2012) The nonlocal reduction method in analyzing the stability of differential equations of induction machines. Doklady Math 85(3):375–379

    Article  MATH  Google Scholar 

  26. Leonov GA, Solov’eva EP (2012) On a special type of stability of differential equations for induction machines with double squirrel -cage rotor. Vestnik St Petersburg Univ Math 45(3):128–135

    Article  MathSciNet  MATH  Google Scholar 

  27. Leonov GA, Bragin VO, Kuznetsov NV (2010) Algorithm for constructing counterexamples to the Kalman problem. Doklady Math 82(1):540–542, DOI 10.1134/S1064562410040101

    Article  MATH  Google Scholar 

  28. Leonov GA, Vagaitsev VI, Kuznetsov NV (2010) Algorithm for localizing Chua attractors based on the harmonic linearization method. Doklady Math 82(1):693–696, DOI 10.1134/S1064562410040411

    Article  MathSciNet  MATH  Google Scholar 

  29. Leonov GA, Kuznetsov NV, Kuznetsova OA, Seledzhi SM, Vagaitsev VI (2011) Hidden oscillations in dynamical systems. Trans Syst Contl 6(2):54–67

    Google Scholar 

  30. Leonov GA, Kuznetsov NV, Vagaitsev VI (2011) Localization of hidden Chua’s attractors. Phys Lett A 375(23):2230–2233, DOI 10.1016/j. physleta.2011.04.037

    Article  MathSciNet  MATH  Google Scholar 

  31. Leonov GA, Kuznetsov NV, Vagaitsev VI (2012) Hidden attractor in smooth Chua systems. Physica D 241(18):1482–1486, DOI 10.1016/j. physd.2012.05.016

    Article  MathSciNet  MATH  Google Scholar 

  32. Leonov GA, Kuznetsov NV, Yuldahsev MV, Yuldashev RV (2012) Analytical method for computation of phase-detector characteristic. IEEE Trans Circ Syst – II: Express Briefs 59(10):633–647, DOI 10.1109/ TCSII.2012.2213362

    Article  Google Scholar 

  33. Leonov GA, Kuznetsov GV (2013) Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractors in Chua circuits. Int J Bifurcat Chaos 23(1):1–69, DOI 10.1142/S0218127413300024

    MathSciNet  Google Scholar 

  34. Marino R, Tomei P, Verrelli C (2010) Induction motor control design. Springer, The Netherlands

    Book  Google Scholar 

  35. Mihajlović N (2005) Torsional and lateral vibrations in flexible rotor systems with friction. Ph.D. dissertation, Eindhoven University of Technology, Eindhoven, Netherlands

    Google Scholar 

  36. Mihajlovic N, van Veggel A, van de Wouw N, Nijmeijer H (2004) Analysis of friction-induced limit cycling in an experimental drill-string system. J Dyn Syst Meas Control 126(4):709–720

    Google Scholar 

  37. Olsson H (1996) Control systems with friction. Ph.D. thesis, Lund Institute of Technology, Sweden

    Google Scholar 

  38. Popp K, Stelter P (1990) Stick-slip vibrations and chaos. Philosoph Trans R Soc Lond 332: 89–105

    Article  MATH  Google Scholar 

  39. Shokir E (2004) A novel pc program for drill string failure detection and prevention before and while drilling specially in new areas. J Oil Gas Bus (1)

    Google Scholar 

  40. den Steen LV (2005) Suppressing stick-slip-induced drill-string oscillations: a hyper stability approach. Ph.D. dissertation, University of Twente

    Google Scholar 

  41. Yakobovich VA, Leonov GA, Gelig AK (2004) Stability of Stationary Sets in Control Systems with Discontinuous Nonlinearities. World Scientific, Singapore

    Book  Google Scholar 

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Kiseleva, M.A., Kuznetsov, N.V., Leonov, G.A., Neittaanmäki, P. (2014). Drilling Systems: Stability and Hidden Oscillations. In: Machado, J., Baleanu, D., Luo, A. (eds) Discontinuity and Complexity in Nonlinear Physical Systems. Nonlinear Systems and Complexity, vol 6. Springer, Cham. https://doi.org/10.1007/978-3-319-01411-1_15

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  • DOI: https://doi.org/10.1007/978-3-319-01411-1_15

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